In-hand manipulation with multi-fingered hands is a challenging problem that recently became feasible with the advent of deep reinforcement learning methods. While most contributions to the task brought improvements in robustness and generalization, this paper addresses the critical performance measure of the speed at which an in-hand manipulation can be performed. We present reinforcement learning policies that can perform in-hand reorientation significantly faster than previous approaches for the complex setting of goal-conditioned reorientation in SO(3) with permanent force closure and tactile feedback only (i.e., using the hand's torque and position sensors). Moreover, we show how policies can be trained to be speed-adjustable, allowing for setting the average orientation speed of the manipulated object during deployment. To this end, we present suitable and minimalistic reinforcement learning objectives for time-optimal and speed-adjustable in-hand manipulation, as well as an analysis based on extensive experiments in simulation. We also demonstrate the zero-shot transfer of the learned policies to the real DLR-Hand II with a wide range of target speeds and the fastest dextrous in-hand manipulation without visual inputs.